How does the structure factor reflect the characteristics of particle distribution?

Question

The structure factor is defined as follows: $$S(\mathbf k)=\frac{1}{N}\sum_i\sum_je^{-\mathbf k\cdot\mathbf r_{ij}\sqrt{-1}}$$ It is related to the radial distribution function by Fourier transform: $$S(\mathbf k)=1+\rho\int[g(\mathbf r)-1]e^{\mathbf k\cdot\mathbf r\sqrt{-1}}d\mathbf r$$ In the paper of Zaidi (physics of fluids, 2020)：

“Generally, larger values of the structure factor for smaller wave numbers indicate faster interactions between close particle pairs or rapid making and breaking of particle clusters, and vice versa.”

How to understand this sentence through the above two relations?

Answer 1

The citation from the paper does not seem based on any theoretical or experimental evidence. The relation between structure factor and radial distribution function is basically a 3-D Fourier transform. This implies that if the radial distribution function, as acknowledged by the author of the paper,

provides information about the overall particle microstructure in a stationary sense

in no way a simple spatial Fourier transform may introduce

information related to the dynamics of the particle microstructures.

Information on dynamics requires time correlation functions, either in real or in reciprocal space. The time-independent correlation functions embodied in the radial distribution function or in the structure factors are not enough to say anything about dynamics, being related to the time correlation functions at the same time.

The presence of large values of the structure factor at small wavenumber is, in general, a signal of the presence of large clusters, but without dynamic information: a fluid close to its critical point or a two-component glass close to the spinodal decomposition may show large values of $S({\bf k})$ at small wavenumbers even if the corresponding typical times are different by many orders of magnitude.